Theorem nnmord 8599

Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnmord	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem nnmord

Step	Hyp	Ref	Expression
1		nnmordi 8598	. . . . 5 ⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2	1	ex 412	. . . 4 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))))
3	2	impcomd 411	. . 3 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
4	3	3adant1 1130	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
5		ne0i 4307	. . . . . . . 8 ⊢ ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐶 ·o 𝐵) ≠ ∅)
6		nnm0r 8577	. . . . . . . . . 10 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
7		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
8	7	eqeq1d 2732	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·o 𝐵) = ∅ ↔ (∅ ·o 𝐵) = ∅))
9	6, 8	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·o 𝐵) = ∅))
10	9	necon3d 2947	. . . . . . . 8 ⊢ (𝐵 ∈ ω → ((𝐶 ·o 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
11	5, 10	syl5 34	. . . . . . 7 ⊢ (𝐵 ∈ ω → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
12	11	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐶 ≠ ∅))
13		nnord 7853	. . . . . . . 8 ⊢ (𝐶 ∈ ω → Ord 𝐶)
14		ord0eln0 6391	. . . . . . . 8 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ ω → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
16	15	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
17	12, 16	sylibrd 259	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
18	17	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → ∅ ∈ 𝐶))
19		oveq2 7398	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))
20	19	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))
21		nnmordi 8598	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
22	21	3adantl2 1168	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
23	20, 22	orim12d 966	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
24	23	con3d 152	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25		simpl3 1194	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
26		simpl1 1192	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
27		nnmcl 8579	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·o 𝐴) ∈ ω)
28	25, 26, 27	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐴) ∈ ω)
29		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
30		nnmcl 8579	. . . . . . . . 9 ⊢ ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·o 𝐵) ∈ ω)
31	25, 29, 30	syl2anc 584	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·o 𝐵) ∈ ω)
32		nnord 7853	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐴) ∈ ω → Ord (𝐶 ·o 𝐴))
33		nnord 7853	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐵) ∈ ω → Ord (𝐶 ·o 𝐵))
34		ordtri2 6370	. . . . . . . . 9 ⊢ ((Ord (𝐶 ·o 𝐴) ∧ Ord (𝐶 ·o 𝐵)) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
35	32, 33, 34	syl2an 596	. . . . . . . 8 ⊢ (((𝐶 ·o 𝐴) ∈ ω ∧ (𝐶 ·o 𝐵) ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
36	28, 31, 35	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
37		nnord 7853	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → Ord 𝐴)
38		nnord 7853	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → Ord 𝐵)
39		ordtri2 6370	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
40	37, 38, 39	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
41	26, 29, 40	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
42	24, 36, 41	3imtr4d 294	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
43	42	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵)))
44	43	com23 86	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (∅ ∈ 𝐶 → 𝐴 ∈ 𝐵)))
45	18, 44	mpdd 43	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
46	45, 18	jcad 512	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → (𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶)))
47	4, 46	impbid 212	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299  Ord word 6334  (class class class)co 7390  ωcom 7845   ·_o comu 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441  df-omul 8442

This theorem is referenced by:  nnmword  8600  nnneo  8622  ltmpi  10864